Steve Bruce's men were second best for long periods and were indebted to goalkeeper Keiren Westwood, who made a string of excellent stops in the second half to frustrate the Potters.
The visitors were forced into making a late change to their starting line-up after Rolando Aarons picked up an injury in the warm-up. He was replaced by George Boyd, with Jordan Thorniley drafted on to the bench.
Stoke, who were pre-season promotion favourites following a decade in the Premier League but have never been out of mid-table, dominated possession in the early stages. Wingers James McLean and Tom Ince got in behind Wednesday's backline on a couple of occasions but their final ball let them down.
As an attacking force, the Owls threatened sporadically. Barry Bannan dragged a low left foot drive from long range well wide after neatly exchanging passes with Gary Hooper.
Centre-forward Hooper, making his first Wednesday appearance since Boxing Day 2017, showed glimpses of his attacking qualities.
However, clear-cut opportunities were at a premium in a cagey first half, although Adam Reach had to be alert to clear delightful inswinging corner from the lively Bojan.
Nearly 3,000 Wednesdayites packed into the away end and a large proportion of them thought Bruce's side had taken the lead just before the half hour mark after Boyd latched on to Hooper's defence splitting pass and coolly rounded Jack Butland before slotting into an unguarded net. But Boyd's celebrations were cut short as his strike was ruled out for offside.
Stoke came within a whisker of opening the scoring in the 40th minute when full-back Tom Edwards tried his luck from just outside the penalty area. His left foot curler flew narrowly wide of Keiren Westwood's left-hand upright.
Moments later, Sam Hutchinson was yellow carded after a rash challenge on Peter Etebo. Owls captain Tom Lees also joined Hutchinson in the book before half-time after upending Benik Afobe.
It was the Potters who made a strong start to the second half. Ince shrugged off the attentions of Michael Hector before slipping in Bojan but Keiren Westwood produced a superb save in the one-on-one situation to thwart the Spaniard.
Westwood came to Wednesday's rescue yet again just after the hour mark, diving to his left to keep out Afobe's bullet header after a pinpoint cross by Edwards.
Bruce freshened up his forward line, taking off Forestieri and Hooper and throwing on Sam Winnall and Atdhe Nuhiu in an attempt to stir his team into life.
But Stoke continued to hold the upperhand and create the bulk of the goal-scoring chances. Ince's attempt was deflected behind before Owls old boy Danny Batth steered a header off target.
After soaking up the Potters pressure, Wednesday briefly put the hosts under the cosh. Winnall headed wide after Nuhiu made a nuisance of himself in the air before Liam Palmer was expertly foiled by the agility of Butland.
Batth wasted a glorious chance to give Stoke the victory in the closing stages. The central defender somehow headed Ince's delightful delivery wide with the goal at his mercy.
The draw extends the Owls' unbeaten Championship run to 12 matches but they are now three points adrift of sixth-placed Derby County with seven matches remaining.
Stoke: Butland; Edwards, Shawcross, Batth, Martins Indi; Allen, Etebo; Ince, Bojan, McLean (Verlinden 70): Afobe (Diouf 66). Substitutes: Federici, Bauer, Williams, Adam, Woods.
Owls: Westwood; Iorfa, Lees, Hector, Palmer; Reach (Matias 89), Hutchinson, Bannan, Boyd; Forestieri (Winnall 58), Hooper (Nuhiu 68). Substitutes: Dawson, Fox, Pelupessy, Thorniley.
Attendance: 26,398 (2,890)
Referee: Andrew Madley (West Yorkshire)